Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations for their number. In this paper we generalize this construction and define tiered trees by allowing more than two classes of vertices. Tiered trees arise naturally when counting the absolutely indecomposable representations of certain quivers, and also when one enumerates torus orbits on certain homogeneous varieties. We define a notion of weight for tiered trees and prove bijections between various weight 0 tiered trees and other combinatorial objects; in particular order n weight 0 maxmin trees are naturally in bijection with permutations on n-1 letters. We conclude by using our weight function to define a new q-analogue of the Eulerian numbers.
In [Dugan-Glennon-Gunnells-Steingrimsson-2019], the authors introduce tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers, and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [D'Adderio-Iraci-VandenWyngaerd-Theta-2021], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [Haglund-Remmel-Wilson-2018].
A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous …
We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for the joint distribution of descents, big ascents and successions of permutations. And then, inspired by the work of Chen and Fu on the trivariate second-order Eulerian polynomials, we show the e-positivity of the multivariate k-th order Eulerian polynomials, which are the enumerators for the joint distribution of ascents, descents and j-plateaux of k-Stirling…
A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with $v>v'$, then $t(v) >t(v')$. For any ordered partition $p=(p_1,p_2,\cdots,p_m)$ of $n$, let $\sett_p$ denote the set of tiered trees with vertex set $\brk{n}$ and with a map $t: \brk{n}\rightarrow \brk{m}$ such that $|t^{-1}(i)|=p_i$ for all $i=1,2,\ldots,m$. For any $T\in \sett_p$, let $K_T$ denote the complete tiered graph whose vertex set and tiering map are t…
Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingrímsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation $σ$ viewed as a sequence of integers, computing the weight of $σ$ involves recursively counting descents of certain subpermutations of $σ$. Using this weight function, one can define a $q$-analog $E_n(x,q)$ of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials $E_n(x,q)$. First, we show that the coefficients of $…
Ma-Ma-Yeh made a beautiful observation that a transformation of the grammar of Dumont instantly leads to the $γ$-positivity of the Eulerian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the $e$-positivity of the trivariate second-order Eulerian polynomi…
We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $γ…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an arbitrary order in a sequence, are in bijection with permutations. The permutation decomposes canonically, by inserting parentheses, into a tree having as nodes a class of permutations which we call primitive. Primitive permutations can be assembled from very simple data. The data for the trees into which a permutation decomposes can be written in a form si…
Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we come to the realization that the grammatical labels play a role as records of chopped off leaves of the original increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial $γ$-positivity of the Eulearian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels $x$ and $y$ in the Gessel representation of a $k$-Stirling permutation by means of an increasing $(k+1)$-ary tree. More …
The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to establish explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The p…
We give two combinatorial proofs of the fact that the number of loopless digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes the $n$-th Catalan number. We construct a bijection with a set of labeled rooted plane trees and with a set of valid parenthesis arrangements.
We introduce Eulerian maps with blocked edges as a general way to implement statistical matter models on random maps by a modification of intrinsic distances. We show how to code these dressed maps by means of mobiles, i.e. decorated trees with labeled vertices, leading to a closed system of recursion relations for their generating functions. We discuss particular solvable cases in detail, as well as various applications of our method to several statistical systems such as spanning trees on quadrangulations, mutually excluding particles on Eulerian triangulations or the Ising model on quadrang…
Remixed Eulerian numbers are a polynomial $q$-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as $q$-binomial coefficients and Garsia--Remmel's $q$-hit numbers. We study their combinatorics in more depth. As polynomials in $q$, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the per…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Brändén. Additionally, we find $q$-exponential generating functions for $q$-analogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. We also present identities relating several of these descent statistic polynomials to refinements of type B Eulerian polynomials and flag descent polynomials by the number of negative letters of a signed permutation. Our m…
The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Pitman-Stanley polytope, and various generalized associahedra related to wonderful compactifications of De Concini-Procesi. These polytopes are constructed as Minkowski sums of simplices. We calculate their volumes and describe their combinatorial structure. …
We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping relies on the existence of canonical orientations which allow to equip the maps with canonical spanning trees, as proved by Bernardi. Our bijection gives in particular (here in the Eulerian case) a…
For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path, $A_F(q)$ specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of $A_F(q)$ is unimodal and that if $\{T_{n}\}$ is a sequence of trees with $|T_{n}| = n$ and maximal down degree $D_{n} = O(n^{0.5-ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.
We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and colored permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalizations of Eulerian polynomials for colored permutations considered recently by N. Williams and the third author, and others. We also discuss stability-- and negative dependence properties satisfied by the partition functions.
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the Möbius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show t…
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a…
In this paper, we introduce the notion of a grammatical labeling to describe a recursive process of generating combinatorial objects based on a context-free grammar. For example, by labeling the ascents and descents of a Stirling permutation, we obtain a grammar for the second-order Eulerian polynomials. By using the grammar for $0$-$1$-$2$ increasing trees given by Dumont, we obtain a grammatical derivation of the generating function of the André polynomials obtained by Foata and Schützenberger, without solving a differential equation. We also find a grammar for the number $T(n,k)$ of permuta…
We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges…
